Noriyasu MASUMOTO, Waseda University, Okubo, Shinjuku, Tokyo , Japan Hiroshi YAMAKAWA, Waseda University

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1 A Study on Predctve Control Usng a Short-Term Predcton Method Based on Chaos Theory (Predctve Control of Nonlnear Systems Usng Plural Predcted Dsturbance Values) Noryasu MASUMOTO, Waseda Unversty, Okubo, Shnjuku, Tokyo , Japan Hrosh YAMAKAWA, Waseda Unversty A predctve control method for nonlnear mechancal systems s dscussed n ths study. values near future of dsturbance tme seres of the controlled system are predcted by a short-term predcton method based on chaos theory whch was proposed by the authors. The deas of the mum samplng perod and the forward horzon are ntroduced n the predcton method and methods to obtan them are shown n ths work. In order to show the effectveness of the proposed method when t s appled to nonlnear systems, numercal examples for a Duffng system are presented. Key Words : Vbraton Control, Predctve Control, Chaos Theory, Short-Term Predcton, Duffng System Duffng mx+ cx+ k 1 x + k 3 x 3 = a cos ω t Duffng a 5001 Duffng,, 64-65, C (1998), Thompson, J.M.T. and Stewart, H.B., Nonlnear Dynamcs and Chaos, (1986), 3, Wley. t * Forward Horzon h f Ke * Backward Horzon h b Dsplacement x m Uncontrolled Predctve control Predctve control + feedback control Tme step Fg. A1 Control results of the dsplacement

values near future of dsturbance tme seres of the controlled system are predcted by a short-term predcton method based on chaos theory whch was proposed by the authors.

2 Duffng Dsturbance Mechancal system Output {W} = (..., w,...) {Y} = (..., y,...) Fg. 1 Dsturbance-output relaton of a mechancal system Table 1 Classfcaton of predctve control methods Mechancal system (Model s known) Lnear Nonlnear Target tme seres of predcton Dsturbance Output Dsturbance Output The number of predcted values used for control forces n x (t)= f x(t) + B u (t)+cw(t) y (t)=dx(t) x y u w n m l f n B C D n 1 n l n m f B C D u u (t)=u p (t)+u f (t) u p u f u p h f 1 u p (k τ)= Ω T w ((k + ) τ) =0 h ^ f w h f Ω k τ τ τ h f h f Ω ( = 0, 1,..., h f -1) h 1 u p ( τ; h)= Ω T w (( + ) τ) =0 h h Ω ( = 0, 1,..., h -1) h o + J h f (τ; h)= x T (( + ) τ) Qx(( + ) τ) =1 h o 1 =0 Ru p (( + ) τ ; h) Q n n R h o

Dsturbance Output Dsturbance Output The number of predcted values used for control forces n x (t)= f x(t) + B u (t)+cw(t) y (t)=dx(t) x y u w n m l f n B C D n 1 n l n m f B C D

3 x ( τ ) = 0 Runge-Kutta τ h J h f ( τ ; h ) h h = 1,,..., h max h h - J h f ( τ ; h ) = 1, 1+1,..., τ h - J h f ( τ ; h ) 1 h - J h f ( τ ; h ) h h f h f h f Ω ( = 0, 1,..., h f -1) = + h o -1 + h o -1 ( + h o -1) + h max -1 h o h max {X} { X}=, x 1, x, x +1, {X} { X * }= x 1, x,, x n η η x ^ n x n +1 δ {X} η δ +1 δ δ η Takens {X * } ^x n + λ ( λ = 1,,..., h f ) {X * } λ {X * } {X * λ} s λ {X * } s λ = mod ( n 1, λ )+1 {X * ^ λ} x n + λ λ {X * } ^x n + λ ( λ = 1,,..., h f )... {X * }=( x 1 x... x s x n -... x n -... x n -1 x n ) δ η η Gona {X * λ} {X * } (η r ) = x (η 1),, x 1, x T ( = η, η +1,, n ) η η r (η ) {X * } η η r (η ) x x x +1 x +1 = f (η ) (η r ) r (η ) f (η ) (η ) x +1 x j +1 (ε)= MAX (η ) (η ) (η r j B (ε) d r ) (η ), r j C B (η ) (ε ) r (η ) ε η d C (η ) (ε)=max x s... x n - x n - {X * }=( x s... x n - x n - x n ) Fg. Generaton of partal tme seres (η ) C (ε) C (η ) (ε ) C (η ) (ε ) η η =, 3, 4,... C (η ) (ε ) η {X * } {X * λ} Ke λ λ 1 λ h f h f h f η η Takens x n

$.., h f ) {X * } λ {X * } {X * λ} s λ {X * } s λ = mod ( n 1, λ )+1 {X * ^ λ} x n + λ λ {X * } ^x n + λ ( λ = 1,,..., h f )... {X * }=( x 1 x... x s......... x n -.$

4 Ke * = MAX Ke λ λ Ke * λ {X * λ} = n η = Ke * * ) C n (ε)= x n +λ x j +λ MAX * ) (ε) d r * ) j r j * ) B n * ) r * ) T = x * 1) λ,, x λ, x ( = s λ + * 1) λ,, n λ, n ) {X * λ} x n ^x n +λ r * ) j r * ) n r * ) n C * ) x (ε)= n +λ x n +λ d r * ) * ) n ^ C p C * ) (ε ) x n +λ x n +λ = x n +λ + C p d ( r * ) * ) n ) when xn x n +λ x n +λ C p d ( r * ) * ) n ) when xn > x n +λ x n x n +λ x n x n +λ {X * } ^x n + λ ( λ = 1,,..., h f ) C p h b h b C p E (µ,c 1 p )= = 1 x +λ (µ,c p ) x +λ x +λ 1 1 n - λ µ Ke * ~ x +λ ( µ, C p ) Ke* µ µ J hb ( µ, C p ) µ Table Optmum desgn problem for C p Desgn varable C p (Intal value) (1.00) Objectve functon J h b (, C p ) = ~ x + (, C p ) - x + x + Constrant condton C p C p 1 C p µ C p µ µ =, 3,... µ-e 1 ( µ, C p ) µ I mn = { µ MIN µ E ( µ,c 1 p )} I mn µ * h b = Ke * + µ * h f +1 µ * C p C p ^x n + λ ( λ = 1,,..., h f ) h b x n -(h b -1),..., x n -1, x n h b µ * C p x n Ke * r * ) n r * ) n µ * r * ) n r * ) n + {X λ * } t Ke * r * ) n - λ r * ) n t t t * t * f ( t ; Ke * )= = 1 * ) g * Ke r + λ * ) g * Ke r + λ g Ke * ( r * ) ) = x gke* r * ) Ke * r * ) ( 1 ) t t ( = 1,,... ) r * ) r * ) Ke * -dmensonal phase space r *) r *) n n + r *) n - r *) r *) n n + - λ Fg. 3 The nearest pont to the current pont n phase space

${X * } ^x n + λ ( λ = 1,,.$

5 r * ) t * Duffng mx+ cx+ k 1 x + k 3 x 3 = a cos ω t a a = 7.50 N Dynamcs a (t) = sn ( t + π 4 ) 6.90 a N f (t)=a (t) cos ω t + nose Dsplacement x m Ampltude a N Ampltude of exctaton a N Fg. 4 Bfurcaton dagram of a Duffng system n ths study Table 3 Parameters of the Duffng system Mass of partcle m Kg 1.00 Dampng coeffcent c Ns/m Stffness coeffcent of x k 1 N/m Stffness coeffcent of x 3 k 3 N/m Angular frequency of exctaton rad/s Tme t sec Fg. 5 Tme seres of dsturbance ( t = sec ) Estmaton value Objectve functon Ke = 5 Ke = 7 Ke = Samplng perod t sec 10 - Fg. 6 Calculaton results of the mum samplng perod Ke Horzon parameter h Fg. 7 h - J h ( τ ; h) plots f Table 4 Embeddng dmensons t * t = sec t = sec t * = sec h - J h f ( τ ; h ) h f = 10 λ = 1,,..., h f = 10 λ Ke * = 9 µ-e 1 ( µ, C p ) µ-c p 349 µ 435 C p µ * = 400 µ * C p = h f =

00 Table 3 Parameters of the Duffng system Mass of partcle m Kg 1.00 Dampng coeffcent c Ns/m 5.00 10 - Stffness coeffcent of x k 1 N/m Stffness coeffcent of x 3 k 3 N/m 3 1.

6 Error rate C p I mn = { } } The number of Ke * -dmensonal ponts Fg. 8 - E (, C p ) plots 1 Dsplacement x m Dsplacement x m The number of Ke * -dmensonal ponts Fg. 9 - C p plots Uncontrolled Predctve control Predctve control + feedback control Tme step Fg. 10 Control results (controlled from the 5001st step) Uncontrolled Predctve control Predctve control + feedback control Tme step Fg. 11 Control results (controlled from the 8001st step) Duffng,,, 11-6, (1975), Farmer, J. D. and Sdorowch, J., Predctng chaotc tme seres, Phys. Rev. Lett., 59-8 (1987), Casdagl, M., Nonlnear predcton of chaotc tme seres, Physca D, 35 (1989), Sughara, G. and May, R. M., Nonlnear forecastng as a way of dstngushng chaos from measurement error n tme seres, Nature, 344 (1990), Gona, M., Cmagall, V., Morgav, G. and Perrone, A., Local predcton of Chaotc tme seres, Proc. the 33rd Mdwest Symposum on Crcuts and Systems, (1990), ,,,, No I (1997-3), Masumoto, N. and Yamakawa, H., A Study on Predctve Control Usng Short-Term Predcton Method Based on Chaos Theory, Proc. the 1997 ASME Internatonal Mechancal Engneerng Congress and Exposton, DE- 95/AMD-3, (1997), ,,,, No I (1998-3), ,,, 64-65, C (1998), Masumoto, N. and Yamakawa, H., A Study on Predctve Control Usng Short-Term Predcton Method Based on Chaos Theory (Determnaton of the Control Force Usng Plural Predcted Dsturbances), Proc. the 1998 ASME Internatonal Mechancal Engneerng Congress and Exposton, DE-97/DSC-65, (1998), ,,, No ( ), Takens, F., Detectng strange attractors n turbulence, Dynamcal Systems and Turbulence, Warwck 1980, Lecture Notes n Mathematcs, 898, (1981), , Sprnger-Verlag. Thompson, J.M.T. and Stewart, H.B., Nonlnear Dynamcs and Chaos, (1986), 3, Wley. Nusse, H.E. and Yorke, J.A., Dynamcs: Numercal Exploratons, nd Ed., (1998), Sprnger-Verlag.

10 Control results (controlled from the 5001st step) 4.00 - Uncontrolled Predctve control Predctve control + feedback control -4.00 8000 8100 800 8300 8400 Tme step Fg.

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